Collaborative Research: Mechanics of Knots and Tangles of Elastic Rods

从航运到航海再到外科手术，每天都要使用数千种不同类型的绳结，每种绳结都需要一定的拧紧力。这种力取决于材料特性、摩擦力和结的拓扑结构。绳结通常也具有承重能力；超过这个水平的力会导致结松开，过大的力可能会导致结中的材料失效。此外，有些结在没有外力的情况下就能紧紧地打牢，而另一些结则很容易解开。换句话说，这些结可以在材料中储存能量。这一概念被称为拓扑电池，可以应用于DNA中的纳米级结和结构工程中的宏观结。该奖项支持了解结的基础科学的研究。这项工作将为结和缠结的力学分析发展建模和计算方法。同时，它将制定实验技术来系统地研究这一力学。该研究将通过开发本科和研究生课程的教学工具（视频、笔记和演示）加以补充。计算机软件也将向公众开放。本项目的研究目的是量化弹性杆中结的力学响应。该项目将采用(1)受计算机图形启发的快速数值模拟，(2)具有定制摩擦的创新材料，以及(3)自主机器人实验来解开结的力学。即使是最基本的打结（手打结），打结所需的力也取决于(1)弹性、(2)摩擦和(3)拓扑结构的复杂相互作用。有趣的是，在拉力超过临界量时，上扣可能会发生断裂屈曲不稳定。这种基本结的不稳定性表明了结力学行为的丰富性。在开发了仿真和实验工具后，将对几种常见结（如手结和鞋带结）的力学响应和不稳定性进行研究。利用仿真工具的计算速度和机器人实验的自主性，将几种类型的结的力学响应量化，以建立其力学库。这些数据将被用来合理化一个结的力学响应的变化作为一个函数的拓扑，材料，和摩擦参数。与元素周期表类似，将制定基于力学的结分类方案，其中结将分为不同的类别，例如摩擦为主的结，弯曲为主的结等。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In settings from shipping to sailing to surgery, thousands of different types of knots are used every day, each requiring a specific amount of force to tighten. This force depends on the material properties, friction, and the topology of the knot. A knot typically also has a load-bearing capacity; force beyond this level causes the knot to be undone, and excessive force may result in material failure in the knot. Moreover, some knots hold tight without any external force while others easily get untangled. In other words, the knots can store energy in the material. This concept is called a topological battery with implications in nanometer-sized knots in DNA to macroscopic knots in structural engineering. This award supports research to understand the fundamental science of knots. The work will develop modeling and computational methods for the analysis of the mechanics of knots and tangles. In parallel, it will formulate experimental techniques to systematically study this mechanics. The research will be complemented by developing teaching tools (videos, notes, and demonstrations) for undergraduate and graduate courses. The computational software will also be made publicly available. The research objective of this project is to quantify the mechanical response of knots tied in elastic rods. The project will employ (1) fast numerical simulations inspired by computer graphics, (2) innovative materials with customizable friction, and (3) autonomous robotic experiments to untangle the mechanics of knots. Even in the case of the most basic type of knots (overhand knots), the force required to tie the knot depends on an intricate interplay of (1) elasticity, (2) friction, and (3) topology. Interestingly, the overhand knot may undergo a snap-through buckling instability beyond a critical amount of pull. Such instability in a basic knot points to the richness of the mechanical behavior of knots. After developing simulation and experimental tools, the mechanical response and instabilities of a few common knots, e.g. overhand and shoelace knots, will be investigated. Exploiting the computational speed of the simulation tool and autonomy of robotic experiments, the mechanical response of several types of knots will be quantified to build a library of their mechanics. This data will be used to rationalize the variation of a knot’s mechanical response as a function of the topological, material, and frictional parameters. Similar to the periodic table of elements, a mechanics-based classification scheme of knots will be formulated, where the knots will be grouped into various classes, such as, friction-dominated knots, bending-dominated knots, and others.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.